Scarlet & Grey
Ohio State University
School of Music

Consonance and Dissonance - Tonotopic Theory

Greenwood (1961) was the first to observe a relationship between critical bandwidth and judgements of consonance/dissonance. Specifically, Greenwood (1961) plotted data from Mayer (1894) against the estimated size of the critical band.

Mayer had collected data where listeners were asked to judge the smallest consonant interval between two pure tones. That is, they were asked to judge the minimum frequency difference where no dissonance was perceived. Greenwood pointed out that the dissonance is judged as absent when the distance between pure tones is roughly the size of a critical bandwidth.

In Figure 9 of Plomp & Levelt's 1964 article, Mayer's 1894 data is again plotted against the critical bandwidth. In this case, Plomp & Levelt used an estimate of the critical bandwidth given by Zwicker, Flottorp & Stevens (1957). There are some discrepancies evident in both the higher and lower frequency ranges. However, the size of the critical band proposed by Zwicker et al is now considered excessively large -- especially in the bass region. A better estimate of the size of critical bands is given in Moore & Glasberg (1983). However, it is now thought that Greenwood's original 1961 estimate of the critical bandwidth is slightly better than even Moore and Glasberg's ERB estimates.

There are a number of questions arising from Mayer's 1894 data. Mayer used tuning forks to generate his stimuli. This raises questions of the purity of the purported pure tones. In addition, Mayer's lowest tuning fork had a frequency of 256 Hz. Mayer asked a friend, R. König, to carry out some additional measures at low frequencies. Mayer's data was collected from 12 subjects, whereas König's data was collected from a single subject.

In 1968, Plomp and Steeneken reported on a modern replication of Mayer's (1894) measures. They collected data from 20 listeners. Subjects adjusted the frequency of the higher of two sine-wave oscillators. In one condition, subjects were asked to adjust the higher tone as close as possible to the fixed tone so that "the two tones did not interfere and could be heard separately." The graph below plots the median values against Greenwood's equation estimating the width of critical bands.

Dissonance & Critical Bandwidth

If the tonotopic theory of sensory dissonance is correct, then one would predict that intervals formed by pitches presented to alternate ears would evoke no dissonance. Sandig (1939) found that intervals formed by playing each tone to separate ears results in a more "neutral" sounding interval.

Hutchinson and Knopoff

William Hutchinson and Leon Knopoff (UCLA Music Dept.) published two papers -- 1978, 1979 on consonance and dissonance. Their goal was to generate numerical tables estimating the perceived dissonance for typical tone pairs (1978) and triads (1979).

The first thing to note about Hutchinson and Knopoff is that they were completely unaware of the work of Kameoka and Kuriyagawa (1969a,b).

Hutchinson and Knopoff begin with a fundamental misconception of Plomp and Levelt. They regard their own work as "an extension of the Helmholtz-Plomp and Levelt model of beating as the cause of dissonance." (1978, p.1)

They misinterpret Plomp and Levelt as follows: "Of the many extensions of Helmholtz's research, perhaps the most recent and comprehensive is that of Plomp and Levelt (1965). These authors reject a variety of other descriptions of consonance and dissonance and reaffirm that the absence of rapid beating is the physical correlate of Western common practice consonance." (1978, p.2)

There is nothing in the Plomp and Levelt approach that takes into account beating.

Another misunderstanding arises with respect to the origin of the critical band. The critical band was posited by S.S. Stevens, and measured by Zwicker.

"By methods of psychological testing, Plomp and Levelt have determined that the critical bandwidth, within which one hears dissonances, is not a constant fraction of the mean frequency of the two tones. Instead, this fraction is smaller for high mean frequencies and larger for lower mean frequencies." (1978, pp.4-5)

In calculating the dissonance for diads and triads, Hutchinson and Knopoff miss an opportunity to note that virtually all triads are more dissonant than virtually all diads. They probably didn't discover this because their equations normalize the dissonance values.

"As a matter of computational convenience, we have "fudged" the frequencies of the overtones of any fundamental to the well-tempered scale." (1978, p.7) Fortunately, they avoid the Zwicker CBW curve and use the data from Cross and Goodwin, and from Mayer. They fit their own curve to the CBW as follows:

CBW = 1.72 (f)^0.65
(compare with Greenwood) They generate dissonance values for various dyads using their equation. They use complex tones consisting of 10 equally-tempered harmonics.

In Hutchinson and Knopoff (1979), the authors turn their attention to triads. They show that a first inversion chord is more dissonant than a root position chord (having the same average pitch). The second inversion chord is slightly more consonant than the root position chord. (Terhardt might have an explanation for why the 2nd inversion chord sounds less "good".)

"All other things being equal, the acoustic rank ordering from the most consonant to the most dissonant for a major triad is: second inversion, root position and firt inversion; for the minor triad it is: first inversion, root position, and second inversion." (1979, p.9) Notice that these orderings are simply a consequence of larger critical bandwidths for lower frequencies. In general, wider intervals between the lowest notes of a chord will generate less sensory dissonance.

Simpson (1994)

Further evidence in support of the tonotopic theory of sensory dissonance is found in the work of Jasba Simpson (1994). Simpson made use of contemporary computer models of the operation of the cochlea. Into these models, Simpson input the stimuli used in five perceptual experiments where listeners judged the degree of consonance or dissonance. Simpson then explored the outputs of the cochlear models to determine whether there were any neurophysiological responses that correlated with the consonance/dissonance judgments. Simpson found that the squared distance of the maximums and minimums from the mean of the maximum and minimums accounted for 58% of the variance in the stimuli used in the five experiments. In effect, the squared distance measure used by Simpson amounts to a measure of tonotopic spread.

Simpson concluded that his findings suggest that dissonance cues are available at the periphery of the auditory system -- in keeping with other extant experimental literature (Sandig, 1939).

Simpson's complete thesis is available online.

An implementation of the Kameoka and Kuriyagawa method for estimating sensory dissonance for any arbitrary spectrum is available.